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The problem asks us to determine \( \frac{dy}{dt} \) in terms of \( x \), \( y \), and \( \frac{dx}{dt} \) using implicit differentiation. We are given the relationship:

\[
y^2 = 4 + xy
\]

Here's how we can approach this using implicit differentiation with respect to \( t \).

### Step 1: Differentiate Both Sides with Respect to \( t \)

1. Differentiate the left side:
   \[
   \frac{d}{dt}(y^2) = 2y \cdot \frac{dy}{dt}
   \]

2. Differentiate the right side:
   \[
   \frac{d}{dt}(4 + xy) = 0 + \left( x \cdot \frac{dy}{dt} + y \cdot \frac{dx}{dt} \right)
   \]
   where we used the product rule for \( xy \) since both \( x \) and \( y \) are functions of \( t \).

### Step 2: Set Up the Equation

Now, we have the differentiated equation:
\[
2y \cdot \frac{dy}{dt} = x \cdot \frac{dy}{dt} + y \cdot \frac{dx}{dt}
\]

### Step 3: Solve for \( \frac{dy}{dt} \)

To isolate \( \frac{dy}{dt} \), bring all terms involving \( \frac{dy}{dt} \) to one side:
\[
2y \cdot \frac{dy}{dt} - x \cdot \frac{dy}{dt} = y \cdot \frac{dx}{dt}
\]

Factor out \( \frac{dy}{dt} \):
\[
\frac{dy}{dt} (2y - x) = y \cdot \frac{dx}{dt}
\]

Finally, solve for \( \frac{dy}{dt} \):
\[
\frac{dy}{dt} = \frac{y \cdot \frac{dx}{dt}}{2y - x}
\]

### Final Answer

\[
\frac{dy}{dt} = \frac{y \cdot \frac{dx}{dt}}{2y - x}
\]

Would you like more details on any step? 

Here are five related questions:
1. How does implicit differentiation differ from explicit differentiation?
2. What is the product rule, and how is it applied here?
3. How do we interpret \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \) in terms of rates of change?
4. How would this process change if the equation were \( y^2 = 4 + x^2y \)?
5. Why is it necessary to bring all \( \frac{dy}{dt} \) terms to one side of the equation?

**Tip:** When solving implicit differentiation problems, keep all terms involving the derivative on one side for easier factorization.

Suppose that x and y are functions of another variable t, and x and y are related by the equation y^2 = 4 + xy. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.

Learn how to find dy/dt using implicit differentiation for the equation y^2 = 4 + xy, given that x and y are functions of t. This detailed solution covers related rates, the product rule, and implicit differentiation, ideal for students in Grades 11-12.

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